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Portions not contributed by visitors are Copyright 2018 Tangient LLC
TES: The largest network of teachers in the world
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Table of Contents
Basics
The 24edo system divides the octave into 24 equal parts of exactly 50 cents each. It is also known as quarter-tone tuning, since it evenly divides the 12-tone equal tempered semitone in two. Quarter-tones are the most commonly used microtonal tuning due to its retention of the familiar 12 tones and since it is the smallest microtonal equal temperament that contains all the 12 notes, and also because of its use in theory and occasionally in practice in Arabic music. It is easy to jump into this tuning and make microtonal music right away using common 12 equal software and even instruments - see this page.24edo as a temperament
The 5-limit approximations in 24-tone equal temperament are the same as those in 12-tone equal temperament, therefore 24-tone equal temperament offers nothing new as far as approximating the 5-limit is concerned. The 7th harmonic-based intervals (7:4, 7:5 and 7:6) are almost as bad in 24-tET as in 12-tET. To achieve a satisfactory level of approximation while maintaining the 12 notes of 12-tET requires high-degree tunings like 36-tET, 72-tET, 84-tET or 156-tET.The tunings supplied by 72 cannot be used for all low-limit just intervals, but they can be used on the 17-limit 3*24 subgroup 2.3.125.35.11.325.17 just intonation subgroup, making some of the excellent approximations of 72 available in 24edo. Chords based on this subgroup afford considerable scope for harmony, including in particular intervals and chords using only 2, 3, 11 and 17. Another approach would be to treat 24-EDO as a 2.3.11.17.19 subgroup temperament, on which it is quite accurate.
Combining ups and downs notation with color notation, qualities can be loosely associated with colors:
The 11th harmonic, and most intervals derived from it, (11:10, 11:9, 11:8, 11:6, 12:11, 15:11, 16:11, 18:11, 20:11) are very well approximated in 24-tone equal temperament. The 24-tone interval of 550 cents is 1.3 cents flatter than 11:8 and is almost indistinguishable from it. In addition, the interval approximating 11:9 is 7 steps which is exactly half the perfect fifth. Some good chords in 24-tET are (the numbers are edosteps, e.g. 4 is a major second, 8 is a major third):
0-4-8-11-14 ("major" chord with a 9:8 and a 11:8 above the root)
Its inversion, 0-3-6-10-14 ("minor")
0-7-14 ("neutral")
0-5-10 (another kind of "neutral", splitting the fourth in two. The 0-5-10 can be extended into a pentatonic scale, 0-5-10-14-19-24 (godzilla), that is close to equi-pentatonic and also close to several Indonesian slêndros. In a similar way 0-7-14 extends to 0-4-7-11-14-18-21-24 (mohajira), a heptatonic scale close to several Arabic scales.)
Commas
24 EDO tempers out the following commas. (Note: This assumes val < 24 38 56 67 83 89 |.)Intervals
24 EDO breaks intervals into two sets of five cartegories. Infra - Minor - Neutral - Major - Ultra for seconds, thirds, sixths, and sevenths; and diminished - narrow - perfect - wide - augmented for fourths, fifths, unison, and octave. For other strange enharmonics, wide and narrow can be used in conjunction with augmented and diminished intervals such as 550 cents being called a narrow diminished fifth and 850 cents being called a wide augmented fifth.
These are the intervals of 24 EDO that do not exist in 12 EDO: 2
See full article on 24 Edo intervals.
infra second, wide unison
35/34
34/33
33/32
50.184
51.682
53.273
large 17-limit quartertone
small 17-limit quartertone
33rd harmonic
infra third
15/13
52/45
247.741
250.304
..
..
27/22
16/13
354.547
359.472
..
tridecimal neutral third
35/27
13/10
449.275
454.214
..
tridecimal subfourth
54/35
17/11
750.725
753.637
..
17-limit subminor sixth
44/27
18/11
845.453
852.592
..
undecimal neutral sixth
26/15
125/72
952.259
955.031
..
..
33/17
35/18
1148.318
1151.230
..
..
Notation
There have been disputes about disadvantages of various systems for notating quarter tones. Here are some of the few systems along with pros and cons.Mainstream Quartertone Notation
Pros: Familiar, fairly easy to learn
Cons: Clutters a score easily, can get confusing when sight read at faster paces
Alternate Quartertone Accidentals
††† (the horizontal line should connect all three vertical lines) = three quarter-tones sharp or Jump-Sharp
For example, the scale 0-5-10-15-20 is written as C-D
Pros: Very easy to distinguish accidentals from one another
Cons: Not practical, tends to clutter a score
And in Persian music
Pros: Easy to read
Cons: Hard to write on a computer, doesn't fit with standard notation well
Sagittal Notation
Sagittal notation works extremely well for 24 Edo notation as well as other systems.It's easy on the eyes, easy to recognize the various symbols and keeps a score looking tidy and neat.
A possibility for the best approach would be to not use traditional sharps and flats altogether and replace them
with Sagittal signs for sharp and flat.
Interval Alterations
The special alterations of the intervals and chords of 12 equal can be notated like this:Supermajor or "Tendo" is a major interval raised a quarter tone
Subminor or "Arto" is a minor interval lowered a quarter tone
Neutral are intervals that exist between the major and minor version of an interval
The prefix under indicates a perfect interval lowered by one quarter tone
The prefix over indicates a perfect interval raised by a quarter tone
The Latin words "tendo" (meaning "expand") and "arto" (meaning "contract") can be used to replace the words "supermajor" and "subminor" in order to shorten the names of the intervals.
Chord Structures
24edo features a rich variety on not only new chords, but also alterations that can be used with regular 12 Edo chords. For example, an approximation of the ninth, eleventh, and thirteenth harmonic can be added to a major triad to create a sort of super-extended chord structure of a major chord: 4:5:6:9:11:13.As for entirely new chords, 24edo features many possibilities for chords. The most obvious is the neutral or mid triad 0-7-14 however there are other options such as
0-9-14 (Ultra Triad or upmajor triad) and 0-5-14 (Infra Triad or downminor triad), the chord names being based on what kind of third is in the chord.
These chords though tend to lack the forcefulness to sound like resolved, tonal sonorities but can be resolved of that issue by using tetrads in place of triads.
For example, the neutral triad can have the neutral 7th added to it to make a full neutral tetrad: 0-7-14-21. However, another option is to replace the neutral third with an 11/8 to produce a sort of 11 limit neutral tetrad. 0-14-21-35 William Lynch considers this chord to be the most consonant tetrad in 24edo involving a neutral tonality. 24 edo also is very good at 15 limit and does 13 quite well allowing barbodos 10:13:15 and barbodos minor triad 26:30:39 to be used as an entirely new harmonic system.
William Lynch considers these as some possible good tetrads:
Due to convenience, the names Arto and tendo have been changed to Ultra and Infra.
Naming Chords in 24edo
Naming chords in 24edo can be achieved by adding a few things to the already [[#|existing]] set of terms that are used to name 12edo chords.They are:
Super + perfect interval such as "perfect fifth" means to raise it by a quarter tone
Sub + perfect interval means to lower a quarter tone
Sharp is to raise by one half tone
Flat is to raise by a half tone
Neutral, arto and tendo refer to triads or tetrads
Neutral, arto, or tendo + interval name of 2nd, 3rd, 6th, or 7th is to alter respectively
Examples:
Neutral Super Eleventh or neut^11 = C neutral 7th chord with a super 11th thrown on top
Arto Sub Seventh Tendo Thirteenth or artsub7^13 = Arto tetrad with an arto seventh and a tendo thirteenth on top Minor Seventh Flat Five Arto Ninth Super Eleventh or m7b5^9^11
Alternatively, ups and downs notation can be used. Here are the blue, green, jade, yellow and red triads:
Rank two temperaments
List of 24et rank two temperaments by badnessList of edo-distinct 24et rank two temperaments
Scales / Modes
Pentatonic:
Hexatonic:
Heptatonic:
Octatonic:
Enneatonic:
Decatonic:
Hendecatonic:
Tridecatonic:
Tetradecatonic:
Instruments
The ever-arising question in microtonal music, how to play it on instruments designed for 12edo, has a relatively simple answer in the case of 24edo: use two standard instruments tuned a quartertone apart. This "12 note octave scales" approach is used in a wide part of the existing literature - see below.Hidekazu Wakabayashi tuned a piano and harp to where the normal sharps and flats are tuned 50 cents higher in which he called Iceface tuning.
Compositions
Music
Microhex3 Microhex4 and Microhex5 by Shaahin MohajeriQuarterpicnic by Chris Vaisvil
Quarter Tone Prelude For Two Harps by Nathan BeDell
Fretless Chrome 1 and Fretless Chrome 2 by Chris Vaisvil
Lament by Jake Freivald. In the freivaldneutral24 scale.
Mo - Happy - Happy play by Jake Freivald in Neuter[7] (2.3.11 mohajira), 24et tuning
Autumn Winds, Easter Time at Nine, Waters of Persia by William Lynch in mohajira, 24et tuning.
Serena, by Mason Green (intro and coda in 24edo, the rest is in 12edo)
Autumn Girl, by Mason Green
About
"Prométhée enchaîné" by Fromental Halévy (considered the first mainstream western orchestral composition to use quarter tones.)"3 Hommages" by Georg Friedrich Haas
List of quartertone pieces on Wikipedia
Practical Theory / Books
External links
quarter-tone / 24-edo - Encyclopedia of Microtonal Music Theory PermalinkAbout 24-EDO by Shaahin Mohajeri Permalink
Notation and Chord Names for 24-EDO by William Lynch
The place of QUARTERTONES in Today's Xenharmonics by Ivor Darreg Permalink